In this post, we are going to continue analyzing the book, How I Wish I’d Taught Maths by Craig Barton about evidence-based education. Today, we will focus on problem solving.
What is a problem?
This is the first question we must answer before addressing the issue.
Let’s see two examples.
- There is a school with 1000 students and 1000 lockers. On the first day of term, the headteacher asks the first student to go along and open every single locker, he asks the second to go to every second locker and close it, the third to go to every third locker and close it if it is open or open it if it is closed, the fourth to go to the fourth locker, and so on. The process is completed with the thousandth student. How many lockers are open at the end?
- In a 20% off sale, a jacket costs $52. What did it cost before the sale?
It seems quite obvious that the first one is a problem, however, is the second?
Schoenfeld (2009) offers two contrasting definitions of ‘word problems’:
- In mathematics, anything required to be done, or requiring the doing of something.
- A question that is perplexing or difficult.
Problem Solving Strategies
Based on John Dewey’s model, which describes stages of thinking in problem-solving, Polya’s early work studies the use of heuristic strategies in solving math problems, proposing a collection of indications, questions, and suggestions to help students in their problem-solving.
These indications are grouped into 4 phases: understanding the problem, developing a strategy, executing the plan, and examining the solution received.
It seems reasonable to assume that to succeed in problem-solving a student must be able to follow these four phases.
However, this assumption deserves closer examination, especially given the difficulty that many students have when going through the stages. In addition, Schoenfeld’s (1992) later work concludes that the use of heuristic strategies is unsuccessful and recommends developing strategies for specific problems.
For example, let’s take a brief look at the second phase: developing a strategy. It’s easy to realize, if we take any math test, that we can’t learn general problem-solving strategies for all the problems that appear. Moreover, we cannot teach such strategies in the absence of specific knowledge of the subject.
Within this second phase, we find tips such as finding a pattern, making a list, drawing, removing possibilities, using a model, considering special cases, working backward or using a formula.
This type of advice, such as drawing, is something that can effectively help students focus on the deep structure of a whole range of problems. However, in order to make a correct diagram, students must be able to incorporate the key characteristics of the problem into the diagram.
Phase Models in Smartick
We also do not work with general heuristic strategies at Smartick. Our problem-solving phase model rests on strategies linked to specific content. We rely on the Singapore method to give specific guidelines on the nature of the problem.
- Schoenfeld, A. 2009. “Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics.” In Handbook for Research on Mathematics Teaching and Learning, 334–70. New York: MacMillan.
- Polya, G. 1945. How to Solve It. Princeton, NJ: Princeton University Press.